Problem 26 Determine for what numbers, if a... [FREE SOLUTION]

Chapter 11: Problem 26

Determine for what numbers, if any, the given function is discontinuous. $$f(x)=c$$

Short Answer

Expert verified

The given function $f(x) = c$ does not have any discontinuity and is continuous for all real numbers.

Step by step solution

Determine the Limit of the Function

A constant function does not change its value for any input. Hence its limit, as $x$ approaches any number, will always be equal to the constant value. Mathematically, if the function $f(x) = c$, then $\lim_{x \to a} f(x) = c$.

Check the Continuity of the Function

A function is continuous at a point $x = a$ if and only if $f(a)$ equals to $\lim_{x \to a} f(x)$. From step 1, we have $\lim_{x \to a} f(x) = c$, and since $f(a)$ will also be $c$ for a constant function, the two values are equal and hence the function is continuous at $x = a$. As this holds true for any arbitrary point $a$, the function is continuous for all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Function

A constant function is a special type of function where the output value remains unchanged for any input value. In mathematical terms, if you have a function defined as $f(x) = c$, where $c$ is a constant, this means no matter what value $x$ takes, the output will always be $c$. Here are some important points to note about constant functions:

The graph of a constant function is a horizontal line on the Cartesian plane that intercepts the y-axis at $c$.
Constant functions are inherently simple and predictable, making them a useful tool in various applications.
For example, if you set room temperature to 22 degrees, it stays constant; that's how a constant function behaves in mathematical terms.

Understanding constant functions helps in comprehending how unchanging values are represented in mathematics.

Limit of a Function

The limit of a function is a fundamental concept in mathematics, often used to understand how a function behaves as its input approaches a certain value. When we talk about the limit of a function $f(x)$ as $x$ approaches $a$, we want to find out what value $f(x)$ gets closer to. This is particularly useful in calculus to explore the behavior of functions at points where they might not be well-defined or might change rapidly.

For a constant function $f(x) = c$, the limit is straightforward. Regardless of how $x$ changes, $f(x)$ remains $c$. Thus, $\lim_{x \to a} f(x) = c$.
Knowing the limit helps determine continuity, which is crucial for understanding the smoothness or breaks in the graph of a function.
Consider limits as a means to smoothly extend the function to a point, even if it doesn't appear to reach a particular value.

Through limits, mathematicians can predict behavior and solve complex problems by looking at nearby points.

Discontinuous Function

A discontinuous function is one where there are abrupt changes or breaks in the function's graph. These breaks occur when a function does not have the same value directly approaching from the left and right side of a particular point. This concept is important in real-life processes where smooth operation is desired. Discontinuities can be visualized as gaps, jumps, or twists in the graph.

A function is continuous at a point $x = a$ if $\lim_{x \to a} f(x)$ equals $f(a)$. If this condition isn't met, the function is said to be discontinuous at that point.
Common types of discontinuities are jump discontinuities, where the function "jumps" from one value to another, and removable discontinuities, where a point is missing or differ from the limit without actually affecting the general behavior.
Discontinuous functions can be found in circuitries, stock price movements, and other practical areas where sudden changes might occur.

In the case of a constant function like $f(x) = c$, there are no discontinuities because the function value never changes. This makes a constant function always continuous.

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91影视

Determine for what numbers, if any, the given function is discontinuous. $$f(x)=c$$

Short Answer

Step by step solution

Determine the Limit of the Function

Check the Continuity of the Function

Key Concepts

Constant Function

Limit of a Function

Discontinuous Function

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